196 PHYSIOLOGIC GENETICS 



Experimental design. — The only technical problem that needs consideration in 

 connection with the degree of genetic determination is the scale of experiment required 

 to estimate it with a given degree of precision. The standard error to be expected 

 when a given number of animals has been measured can be deduced in the following 

 way. 



Let V be the estimate of the variance in the genetically variable group, based on 

 N v degrees of freedom; and let U be the estimate of the variance in the genetically 

 uniform group, based on N u degrees of freedom. For the purposes of planning the 

 degrees of freedom may be taken to be the number of animals measured. The sampling 

 variances of these two estimates of variance are given by 



a\ = 2V 2 IN v and o\ = 2U 2 /N U , (1) 



where a 2 is the sampling variance. The standard error of each estimate is the square 

 root of its sampling variance. The ratio of the standard error to the variance itself is 



a v /V= V'Wv and a v \U = V2[N~ U . (2) 



This ratio is plotted in curve (a) of figure 34. 



The degree of genetic determination, g, is estimated as 



g=(V- U)\V= 1 - U/V (3) 



The sampling variance of the degree of genetic determination (a 2 g ) can be deduced 

 from the general properties of variances as follows : 



n 2 — n 2 



'->g — ° (1- UIV) 



— n 2 



= Y*( v2(y2 v + u2 °'' 



u 2 

 v 2 



Since UjV = 1 — g, from (3), the standard error of the estimate reduces to 



°^^-^HhI) 



(4) 



Now, the total number of animals that can be measured is fixed by the amount of 

 space available, or of effort that can be expended. The formula deduced above for 

 the standard error (4) shows that with a fixed total (that is, N u + N v ), the standard 

 error will be minimal if N u = N v . Therefore the best design for the experiment is 

 to have equal numbers of animals measured from the genetically variable and from the 

 genetically uniform groups. If this number is N (so that a total of 2N animals are 

 measured), then we obtain the relationship 



-*- = — ■ (5) 



This ratio is plotted in curve (b) of figure 34. 



